ARTICLE ISSN 2651 - 5210

MODULAR 2020;3(2):179-194

179

A Geometric Method on Facade Form Design with Voronoi Diagram

Voronoi Diyagramı ile Mimari Cephe Tasarımı Üzerine Geometrik Bir Yöntem

Helin POLAT1, Zeynep Yeşim İLERİSOY2

Received: 03.09.2020 - Accepted: 30.10.2020

Abstract

Voronoi is a stochastic pattern that is the result of structural formation with the least material and least

energy in nature. Therefore, Voronoi, which has become a biomimetic pattern, is a source of inspiration

in architectural design and it has been increasingly used in this field. Modern design methods make it

possible to adapt the process of self-organization of biological structures to architecture using

mathematical models such as Voronoi. Especially on the facade, it is used on randomness and therefore

irregularity as in nature and these irregular forms can be built thanks to the possibilities provided by

technology. However, randomness limits the designer's ability to interfere with the form. In this study, a

method was presented which including modeling process and material usage amount using Rhinoceros

and Grasshopper software. With this method, it was aimed to make Voronoi a tool that the designer can

control the process while producing patterns and to create a regular and systematic design principle by

integrating it with the balance principle of geometry. The patterns whose impact areas were changed with

symmetry, asymmetry and radial balance approaches were evaluated by comparing the amount of

material used and their effect on the application process was evaluated. As a result, it was determined that

similar increases occurred between the level of inclusion of direction and movement in the design and the

level of randomness in the process of determining the most efficient one among alternative patterns in

terms of material usage, as in nature.

Keywords: Architectural Facade Design, Principle of Balance in Geometry, Voronoi Diagram,

Computer-Based Design, Biomimetic Pattern.

Özet

Voronoi doğada en az malzeme ve en az enerji ile yapısal biçimlenmenin bir sonucu olan stokastik bir

desendir. Bu sebeple biyomimetik bir desen olma özelliği kazanan Voronoi, mimari tasarıma ilham

kaynağı olmakta ve bu alandaki kullanımı giderek artmaktadır. Modern tasarım yöntemleri, matematiksel

modeller kullanarak Voronoi gibi biyolojik yapıların özörgütlenme sürecinin mimariye uyarlanmasını

mümkün kılmaktadır. Özellikle cephedeki kullanımı doğada olduğu gibi rastgelelik ve dolayısıyla

düzensizlik üzerine kurulu olmakta ve teknolojinin verdiği imkânlar sayesinde bu düzensiz biçimler inşa

edilebilmektedir. Ancak rastgelelik, tasarımcının biçime müdahale etme durumunu kısıtlamaktadır.

Çalışmada Rhinoceros ve Grasshopper yazılımları kullanılarak modelleme süreci ve malzeme kullanım

miktarı bilgisini içeren bir yöntem sunulmaktadır. Bu yöntem ile Voronoi’yi, tasarımcının desen üretirken

süreci kontrol edebildiği bir araç haline getirmek ve geometrinin denge ilkesi ile bütünleştirerek düzenli

ve sistematik bir tasarım prensibi oluşturmak amaçlanmaktadır. Simetri, asimetri ve radyal denge

yaklaşımları ile etki alanları değiştirilen desenler, kullanılan malzeme miktarına yönelik karşılaştırma ile

ele alınmış ve uygulama sürecine etkisi değerlendirilmiştir. Sonuç olarak, alternatif desenler arasından

malzeme kullanımı bakımından, doğada olduğu gibi, en verimli olanı tespit etme sürecinde yön ve

hareketin tasarıma dahil edilme düzeyi ile rastgelelik düzeyi arasında benzer artışların oluştuğu tespit

edilmiştir.

Anahtar Kelimeler: Cephe Tasarımı, Geometride Denge İlkesi, Voronoi Diyagramları, Bilgisayar

Destekli Tasarım, Biyomimetik Desen.

1 Gazi Üniversitesi, Mimarlık Bölümü, polathelin_04@hotmail.com, ORCID: 0000-0002-4349-2258 2 Gazi Üniversitesi, Mimarlık Bölümü, zyharmankaya@gazi.edu.tr, ORCID: 0000-0003-1903-9119

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1. Introduction

The Voronoi diagram can be found in many places in nature. From cell division to

patterns in animal skin, from the structure of the fly wing to the leaf structure; it can be

seen in many examples in various scales. Formal formations in nature such as Voronoi

are integrated into man-made objects to create more efficient and sustainable structures.

Being inspired by nature today; It investigates not only the direct transfer of form but

also the formation process of the form in nature. In this case, it is possible to define

complex forms and systems of biological structures by using appropriate mathematical

models (Nowak & Rokicki, 2016).

Voronoi Diagrams, which are formed with a set of points and have a polygonal cell

structure, are becoming widespread in architecture today. Many architects use Voronoi

Diagrams, a method of space discretization, to shape structural forms, create patterns for

the facades of buildings, and design spatial forms. Architectural facades created using

Voronoi are designed as an irregular shape, just like in nature. In this case, the design is

created automatically by the computer, and the decision-making authority of the

designer is constrained. Thanks to the method presented in this study, the designer can

use the Voronoi diagram to produce pattern alternatives in line with the own rules

through the balance principle and can determine the most efficient one among these

patterns in terms of material used.

Within this study to create a pattern for the architectural facade, firstly, square, and

circular shaped basic grids were created using Voronoi with periodically arranged point

clusters. The patterns are obtained by changing the positions of the points on these

grids. Grid boundaries and point numbers were kept constant. In order to increase the

visual perception on the facade, it was aimed to have a geometric order of the patterns,

and in this direction, the points are moved as a group on the grid. This grouping was

called the "impact area" in the study. Geometric order was obtained through the

principle of balance. On the grids arranged with three different balance principles as

symmetric, asymmetric, and radial, the affected areas that give points rotation and

translation capabilities are determined. Points outside the impact area have not been

interfered by the designer. This affects the size of the Voronoi cells and thus the shape

of the pattern. The line lengths of the new patterns were obtained to determine the

amount of rod material to be used for facade production. Within the study, the

relationship between the amount of displacement of the points in different types and the

total length of the rod to be used was investigated.

2. Voronoi Diagram

The emergence of the Voronoi diagram, which is a graphical explanation of structure

formation in nature, dates back to the 17th century. It is seen that Descartes used similar

diagrams in his book of principles of philosophy in 1644 to explain that the solar system

consists of vortices. The first comprehensive presentations were featured in the work of

Gustav Dirichlet and Georgy Voronoi. Voronoi still maintains its importance since its

emergence and continues to be developed with new technology in diversified fields.

Today, it is used in many different fields that are involved in science and technology

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such as astrophysics, ecology, geometry, hydrology, meteorology, computer graphics,

and statistics (Okabe et al., 2000).

The mathematical definition of the Voronoi diagram can be briefly expressed as the

process by which a specified set of points divides the entire area into parts. Each point

determines the boundaries of its sub-area within the whole area concerning the location

of neighboring points (Coates et al., 2005). All locations within the boundaries of a

Voronoi cell are closer to the point in the center of the cell than other points

(Aurenhammer, 1991). The diagram, which can be created in both two and three

dimensions (Figure 1), is convex polygons consisting of centers (points), cell edges and

cell vertices in two dimensions; They are polyhedrons consisting of points, vertices and

surfaces in three dimensions (Nowak & Rokicki, 2016).

(a) (b)

Figure 1. (a) Voronoi diagram in 2D (b) and in 3D (Fortune, 2017).

The diagram, which is shaped according to the relationship of each point with its

neighboring points, reorganizes itself when the positions of the points are changed or

when point extraction and addition processes are applied. Thus, it is a complex system

configured by the relationships between points (Coates et al., 2005).

3. The Voronoi Diagram Within the Context of Geometry

Voronoi diagrams created with a random set of points are a product of computational

mathematics. In the field of architectural design, it is a widespread view that uncertainty

and randomness are an integral part of Voronoi. Thereby, it is believed that patterns

similar to the formations in nature are obtained (Wu & Zhang, 2016). However,

Voronoi diagrams do not have to be created only with random points in structural

layout. Any set of points can be used as input to create grids in the plane. In Voronoi

diagrams where the positions of the points are determined by the designer, geometric

cells are obtained by placing the points periodically in the plane. For example, square

cells are formed in the layout where the distance between points is fixed and the points

are in the center of the squares (Figure 2a). Similarly, hexagonal grids are a result of

triangular point placement (Figure 2b). Voronoi diagrams created with randomly placed

points (Figure 2c), on the other hand, have a very soft composition in many respects and

are more variable, asymmetric, scalable, and irregular compared to hard grids such as

square and rectangular.

Grids such as square and hexagon are particular cases of the Voronoi diagram. In

addition to having the ability to imitate other grids, it also can change and transform.

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This feature of manageability allows the designer to switch from one grid to another

during the design process (Kardasis, 2011).

(a) (b) (c)

Figure 2. a and b are patterns created from periodically placed points; (a) Square grids, (b)

Hexagonal grids, (c) Voronoi diagram created from randomly placed points (Kardasis, 2011).

4. Principle of Balance

The physical-mental existence of human and all organic life tend to achieve balance. An

imbalanced composition appears accidental, discontinuous and, therefore useless. The

elements involved in the structure tend to change shape and location at all times to

achieve a state that is better in harmony with the whole. On the contrary, in a balanced

composition, there is a distribution in which all the actions included in the system, in

both physical and visual balance, are stopped. It creates a sense of stability and

robustness against the building in those who observe the building (Arnheim, 1974).

Besides that one, the shapes arranged with the principle of balance have the feature of

being perceived to a great extent by the observer at first sight.

The architect wants to take visual decisions more systematically while designing. In this

case, the balance becomes an important factor in determining the dynamics of the

composition (Arnheim, 1974). There are three types of balance elements: symmetrical,

asymmetrical, and radial. Symmetry occurs with the presence of a center or an axis. The

symmetry axis consisting of two points provides a balanced distribution of the elements

included in the plane (Ching, 2007). In geometry, symmetry is created by rotation,

reflection, and translation. Symmetrical balance is achieved by having the same or very

similar elements on both sides of the area divided by horizontal, vertical, or an axis

located at any angle on the plane. Radial balance provides a visual focal point where

elements are positioned circularly around a center. Unlike symmetry and circular

balance, asymmetry is not arranged around a certain axis. Asymmetry creates a complex

design structure where the elements balance each other and, therefore it is the most

dynamic balance (Zhao et al., 2014).

5. Use of Voronoi in Architectural Applications

The skills such as manual dexterity, mastery, memorization, and rapid calculation

required for traditional construction techniques are no longer sought in today's design

techniques. Digital design tools replace these abilities and require intellectual skills such

as intelligence, thinking and decision making (Terzidis, 2009). However, traditional

techniques are being gradually abandoned in order to pave the way for new design

challenges and to build more efficient structures (Angelucci & Mollaioli, 2018). Digital

design is based on algorithmic designs that have the ability to generate code. It is

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possible to generate the structure of a cellular grid system by changing the values of its

components (Oxman & Oxman, 2010). In this way, the architect can create individual

tools and produce the most appropriate architectural forms using the scripting language.

Creating a structural surface using rod elements based on topological information

provides in terms of architecture interesting designs (Gawell & Nowak, 2015).

Facade applications in architecture such as determining the shape of the panels,

tessellation of the double-sided facade, and structural shaping of the building skin are

observed (Nowak & Rokicki, 2016). Especially with the latest developments in

technology, tall buildings with a diagrid system are shaped with nature-inspired cellular

patterns. These biomimetic patterns, what the degree of irregularity can be adjusted

arbitrarily, are preferred to obtain aesthetically attractive facades (Angelucci &

Mollaioli, 2018). Besides its aesthetic feature, it provides a form with the least material

and least energy, just like in nature (Dimcic, 2011). Melbourne Recital Center in

Melbourne and Alibaba Headquarters in Hangzhou are sample buildings with a facade

shaped by Voronoi (Figure 3) (Nowak, 2015).

(a) (b)

Figure 3. (a)Melbourne Recital Center, (b)Alibaba Headquarters (Url-1, Url-2)

The load transfer of mosaic facades with such complex cell structure is generally as

follows: Some nodes of the structural grid used in the facade are installed at the points

where they intersect with the the load-bearing points of the building. In this case, the

loads on the non-intersecting facade nodes are indirectly transferred to the structural

system of the building (Brzezicki, 2018).

When the literature is examined, experimental studies have been reached including

analysis and research on the shell, structural system, and space design formed with

Voronoi. For example, Tonelli et al (2016) used the Voronoi diagram to design grid

shell structures. The tensor area obtained in the result of the stress analysis on the initial

surface was used as a metric. Voronoi tessellation was performed on this metric, and

then aesthetics were improved by using symmetry and ensuring the regularity of the

cells. Thus, in the study, a method was presented on shell design with strong static

performance and aesthetically pleasing appearance with Voronoi (Tonelli et al., 2016).

Herr and Fischer (2013) researched to find a modern alternative for the triangular and

rectangular beam layout using Voronoi diagrams. The patterns were dynamically

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generated by defining algorithms. The patterns generated for the beam and column

layouts are discussed in terms of density, the capacity of load carrying, and aesthetic

appearance (Herr & Fischer, 2013). Friedrich (2008) researched the static performance

of the 3D Voronoi diagram as a framed structural system. The edges of polyhedrons

generated with Voronoi are considered structural members of a static system. Different

techniques were investigated to optimize the cell structure in terms of its structural

features (Friedrich, 2008). A study was carried out by Harwiansyah (2016), which

includes using 3D Voronoi diagrams for a house design in which trees act as carriers in

a forest area. In the study, trees were taken as a point set. Thus, the settlement of trees

became the factor determining the formal features of the spaces that make up the house

(Harwiansyah, 2016).

Some studies focus on facade design using Voronoi diagrams and analyze grid

structures generated in this way (Torghabehi & Buelow, 2014; Gawell & Nowak, 2015;

Gawell & Rokicki, 2016; Mele et al., 2016). Torghabehi and Buelow (2014) presented a

study proposing to create a Voronoi patterned building skin for a mid-rise building. For

the outer skin, the desired data were introduced to the system such as the complexity of

the Voronoi pattern and then, material efficiency and resistance to environmental effects

and alternatives were produced using the genetic algorithm. Then, the algorithm

presented the best possible results for the designer (Torghabehi & Von Buelow, 2014).

Gawell and Nowak (2015) performed analytical tests on the efficiency of planar rod

structures generated using the Voronoi diagram. The horizontal, vertical, and cross axes

that prevailing in the construction industry became the determining elements in shaping

the patterns. It has been determined that the patterns created in the metric with the same

number of points produce different results in terms of efficiency. As a result of the

study, it was emphasized that different patterns can be produced according to the needs

of the user using Voronoi tessellation (Gawell & Nowak, 2015). In the study of Rokicki

and Gawell (2016), the efficiency of various grid systems developed in the digital

environment in terms of material consumption was discussed. Three different metric

dispositions were created, regular geometry, Voronoi, and Delaunay, and each layout

were analyzed within itself. According to the results obtained from the study, it revealed

that the array of points was a factor affecting the total length of the material used

(Rokicki & Gawell, 2016). In the study conducted by Mele et al., (2016) Productivity

analyzes were carried out for the building skin created using the Voronoi diagram in

high-rise buildings, according to the density and irregularity of the pattern. Efficiency

was obtained by dividing the total weight of the structural steel used for the model by

the total floor area of the building, and it was found that as the irregularity increased, the

structural weight generally decreased (Mele et al., 2016).

It is understood that there is an orientation towards the interpretation of the new shapes

formed by changing the point positions on the diagram according to material efficiency

in the limited studies specific to the facade (Table 1). However, when considered with

the basic design principles, its change in both visual and quantitative aspects will be

important for its evaluation in the design process. It will contribute to the literature on

creating an answer to this question: “Can the Voronoi diagram be designed with the

balance principle, which is one of the basic design principles of geometry?”.

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Table 1. An Evaluation on Voronoi Diagrams Used in Facade Design Authors Purpose Constants Variables Outcome

1

To

rgh

abeh

i an

d

Bu

elo

w, 2

01

4

It was aimed to find the

most efficient patterns

produced using the

Voronoi diagram for

building skin design

with the help of genetic

algorithm (GA).

-The dimension of

the exterior surface

of the building

- Environmental

factors

- Aesthetic value

(complexity level

of the Voronoi

pattern)

The width of

the tubular

system and

the formal

differences of

the Voronoi

pattern

The algorithm has

found the most

effective patterns in

line with the

determined criteria.

2

Gaw

ell

and

No

wak

, 20

15

Different dispositions

of Voronoi patterns

were compared.

- Metric system

- Voronoi points

and number of cell

- Axis direction of

the rods (must be

horizontal or

orthogonal)

The locations

of Voronoi

points on the

metric

It was determined that

the efficiency values

of different Voronoi

patterns obtained by

simply changing the

positions of the points

are different.

3

Ro

kic

ki

and

Gaw

ell,

20

16

Voronoi diagram and

other grid structures are

compared separately on

efficiency.

- Metric system

- Voronoi points

and number of cell

The locations

of Voronoi

points on the

metric

It was concluded that

the layout for the

Voronoi patterned

structure affects the

system weight.

4

Mel

e et

al.

,20

19

The efficiency of the

Voronoi pattern for

tubular system design

in high-rise buildings

was optimized

according to the

disposition and density.

-The dimension of

the exterior surface

of the building

The regularity

and density of

the Voronoi

diagram

It was determined that

as the irregularity of

the Voronoi diagram

increases, the

structural weight

generally decreases.

6. Method and Case Study

The method that enables the designers who create facades using Voronoi to

systematically transform the mathematical principle into form was presented. It was

accepted that the facade created by this method was a self-supporting system created

with structural rods in the vertical plane. The production of the facade was considered to

be done by induction method, in the simplest terms. The cells are formed by combining

the rods, then the cells are combined to complete the form of the facade.

The graphs showing the data of the patterns produced represent the total structural rod

length and the longest rod length to be used on the facade. Thanks to the total structural

rod length data, the designer can determine the optimum during pattern generation. The

longest rod length data will assist in determining the length of the material in parallel

with the strength of the material in the structural analysis process, which will then be

carried out independently. In addition, if more structural rods of the same length were to

be used for the facade, the physical production of the designed facades would be easier

and faster.

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Based on the aforementioned reasons (Section 4), 3 different layouts were created with

the balance principle, which is one of the design principles of geometry. The 3 layouts

were designed with symmetry, asymmetry, and circular balance, respectively. Based

upon the feature of the Voronoi diagram that can be repositioned and transformed by

means of points, the design was carried out through the "Grasshopper" plugin of the

Rhinoceros program.

First, points were placed at equal distances on the rhinoceros x-z plane to form the

pattern. The point (pt) component has been added to the Grasshopper canvas (Figure 4)

and the points set to this component. The points were moved by the designer in separate

directions for each balance principle. Since the distance between the points was

arranged as 6m, the frame enclosing the points was determined 3 m away from the

outermost points. This was because each point in the Voronoi diagram was considered

to be the center of the cell. This limit was set for the boundary input of the Voronoi

component. Then the point input of the Voronoi component was connected to the pt

component to create Voronoi from the points.

Figure 4. Grasshopper script to create and analyze patterns

It was necessary to follow the marked path 1 in Figure 4 to determine the total rod

length. The Voronoi component was connected to the Lenght (Len) component to

determine the length of each line that generates the Voronoi diagram. Then Len was

connected to the Mass Addition (MA) component to determine the total length of the

bar. So the lengths of the rods obtained with the Len were summed. The result of the

calculation was reflected in the panel component.

In order to find the longest bar length, it was necessary to use path 2. Voronoi was

connected to the Explode component after it was connected to the Pt component.

Because the Voronoi component gives the total length of the cells. It was necessary to

use the explode component to detect the lines each cell had separately. It was then

linked to the flatten tree component. Because the line lengths obtained after explode

were in the form of groups. Each Voronoi cell represents a group. However, the group

was not suitable because it was wanted to determine the longest one among all the lines.

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The sort list component was used to create this list and detect the longest line. The result

was reflected in the panel component.

Two different grids were created on the vertical plane. One reference grid was created

for the patterns obtained through symmetry and asymmetric balance, and a different

reference grid was created for patterns based on radial balance. Different impact areas

were determined on these reference grids created with different balance principles

(Table 2). Rotation for symmetrical balance, movement in the determined direction for

asymmetric balance, and translational movements towards the center for radial balance

was applied to the impact area. The amount of displacement has been gradually

increased for each balance principle. The positions of the points included in the impact

area on the grid have changed, and new patterns have been formed at each stage. The

amount of material required for the structure was determined by calculating the total

line length of the patterns obtained.

Table 2. Topologies of İmpact Areas and their Constants and Variables.

Balance

Principle

Symmetry Asymmetry Radial

Constants

The number of points, plane

boundaries, the axis of

symmetry, material properties

used

The number of points, plane

boundaries, material properties

used

The number of points, plane

boundaries, location of the

center on the plane, material

properties used

Variables

The position of points on the

plane at each step relative to

the angle of rotation of the

impact area.

The position of points on the

plane at each step relative to

the direction of movement of

the impact area.

The position of the points on

the plane at each step relative

to the distance of the impact

area to the center.

6.1. Voronoi Diagram Generated with Symmetrical Balance

After the symmetry axes required for symmetrical balance were determined on the

plane, the shape and size of the impact area were defined. Two symmetry axes

intersecting each other at right angles and 4 square-shaped impact areas, each affecting

9 points, were shown schematically in Figure 5.

Figure 5. The positions of the impact areas on the grid were positioned in accordance with the

symmetrical disposition

The reference grid determined as the starting pattern (Figure 6a) was created from 49

points and had a square Cartesian feature. Rod length was determined as 6 m, total rod

length was 1176 m. These figures were used as a reference for comparison with the

patterns created. The impact area determined on this grid is positioned to be

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symmetrical to each other. In each step, the rotation angle of the impact areas on the

plane was increased by 15 degrees and rotated in 3 different angles namely 20, 45 and

60 degrees, respectively (Figure 6).

(a) (b) (c) (d)

Figure 6. Topological Voronoi structure variants generated by rotating at different angles; (a)

Reference Grid(X), (b) A1 grid, (c) A2 grid, (d) A3 grid

The total length of the rod and the longest rod length of each of the 3 different patterns

(Figure 7) obtained were compared. As a result of the analysis, it was determined that

with the increase in the angle value, the total length of the rod to be used to generate the

pattern increased. In addition, as the angle of the 6 m long bars forming the reference

grid increased, the distances between each other changed in relation to the new positions

of the points. And accordingly, it was observed that the length value of the longest rod

in the pattern gradually decreased. It was determined that A1 was the most efficient

pattern in terms of material usage among the patterns obtained with symmetrical

balance layout (Figure 7).

Figure 7. Analysis of properties that change as the angle of rotation of the domain increases in

the radial balance disposition

0° 20° 45° 60°

X A1 A2 A3

TOTAL ROD

LENGHT(m)1176 1177 1184 1199

1170

1180

1190

1200

TOTAL ROD LENGHT

20° 45° 60°

A1 A2 A3

LONGEST ROD

LENGHT(m)10,7 9,5 9

9

9,5

10

10,5

11LONGEST BAR LENGTH

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6.2. Voronoi diagram generated with asymmetrical balance

In the asymmetric balance disposition where there was no specific axis, the location of

the impact areas according to the visual weight was determined. 3 different patterns

were obtained by providing vectorial displacement of the domains placed at different

angles in different directions. In the comparisons, the reference grid created with the

symmetrical balance was taken as the starting pattern with the same properties (Figure

8a).

B1, B2, and B3 patterns were created by moving 1, 2, and 3 units, respectively, in the

directions expressed in Table 1. Voronoi patterns obtained were shown in Figure 8.

(a) (b) (c) (d)

Figure 8. Topological Voronoi structure variants with asymmetrical balance disposition; (a)

Reference Grid(X), (b) B1 grid, (c) B2 grid, (d) B3 grid

When the total rod lengths were compared, no steady increase or decrease was found on

the values as a result of the change in the amount of displacement (Figure 9). The

reason for this was the irregularity of the impact areas located asymmetrically and the

points that were moved in more than one different direction. On the other hand, the

lengths of the 6 m long bars that make up the reference grid changed as the amount of

displacement increased, it was observed that the length gradually increased. Among the

patterns obtained with the asymmetrical balance layout, B2 was found to be the most

efficient pattern in terms of material usage.

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Figure 9. Analysis of structural properties that change as the amount of displacement of the

points increases in the asymmetrical balance disposition

6.3. Voronoi Diagram Created with Circular Balance

For the radial balance in which the elements were circularly positioned around a center,

a different grid was taken as reference from the reference grid used in other layouts. The

value of points of the reference grid that supports radial balance (Figure 10a) and the

reference grid used for symmetry and asymmetric balance were close to each other. The

grid consists of 48 points, the total bar length was 1154 m, the longest bar length was

9.4 m. These figures were taken as reference for comparisons to be made.

The points that construct the diagram have been moved towards the center in the way

shown in Table 2. C1, C2 and C3 grids were the resulting products of the displacement

of the points in the reference grid towards the center by 1, 2 and 3 units, respectively

(Figure 10).

(a) (b) (c) (d)

Figure 10. Topological Voronoi structure variants with radial balance disposition; (a)

Reference Grid(Y), (b) C1 grid, (c) C2 grid, (d) C3 grid

0 unit 1 unit 2 unit 3 unit

X B1 B2 B3

TOTAL ROD

LENGHT(m)1176 1148 1143 1156

1140

1150

1160

1170

1180

TOTAL ROD LENGHT

0 unit 1 unit 2 unit

B1 B2 B3

LONGEST ROD

LENGTS(m)7 8 9

6

7

8

9LONGEST ROD LENGTH

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According to the results, it was observed that as the amount of displacement towards the

center increased, the total length of the rod used increased. In parallel with this, it was

determined that the longest rod length increased with the increase in the distance

between points (Figure 11). The movement of the points caused the patterns to scale

proportionally, and accordingly, the material lengths gradually increased. It was

determined that C1 was the most efficient pattern in terms of material usage among the

patterns obtained with radial balance layout.

Figure 11. Analysis of the structural properties that change as the displacement of the points

towards the center increases in the radial balanced disposition

7. Research results and discussion

Today, Voronoi tessellation is used as a space discretization tool to design roofs,

facades, walls, and similar elements. Modern architectural understanding changes its

modeling methods in parallel with technology and tends towards digital design. Voronoi

diagrams, which can adapt and transform, provide flexibility to the designer in the

architectural design process. In addition, with the help of the code created in the

program, the design process can be improved by making selections thanks to the fact

that all components of the design can be changed separately. Thus, the digital

environment allows the designer to quickly change the properties of the final product at

any time in the pattern generation process and to easily compare with previous data.

Within this research, the balance principle, one of the basic principles of design, was

taken as a basis in order to shape Voronoi diagrams. Rotation, translation, and

movement towards the center were carried out on three different orders created with the

principles of symmetry, asymmetry, and radial balance. These applications, realized by

0 unit 1 unit 2 unit 3 unit

Y C1 C2 C3

TOTAL ROD

LENGHT(m)1154 1164 1174 1184

1150

1160

1170

1180

1190TOTAL ROD LENGHT

1 unit 2 unit 3 unit

C1 C2 C3

LONGEST ROD

LENGHT (m)9,9 10,4 11,3

9

10

11

12LONGEST ROD LENGHT

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192

changing the positions of the points, have been provided with a script produced in the

Grasshopper. As a result, it was determined that the total rod lengths of the patterns

obtained and the longest rod length to be used differ for each balance principle.

When the designer created patterns with symmetrical and radial balance, it was

sufficient to set a single rule for the whole pattern, but he had to specify more than one

direction and movement in asymmetrical balance. For this reason, in diagrams with

symmetrical and radial balance, it was observed that as the displacement amount of

points increases, the amount of material used increases. However, there was no clear

result for asymmetrical balance. In addition, patterns in symmetry and radial balance

had a more regular composition compared to asymmetrical balance. This layout

contributes to the perception of the facade as a whole. On the other hand, if there are

more cells of the same size that make up the pattern, it means that structural rods of the

same length will be used. The manufacturing process is quite complex when there are

many unique Voronoi cells in the pattern generated. For this reason, the physical

production of the facades created with the symmetrical and radial principle would be

easier than the production with the asymmetric principle.

The impact area determination method specified in the study contributes to the designer

to create complex patterns indirectly and to create this pattern in line with the

boundaries determined by the balance principle, not randomly. The designer can obtain

a large number of products with a small number of parameters and can use this method

as a pattern exploration tool. Accessing at the same time, both of the images of the final

product and the amount of material to be used speeds up the designer's decision-making

process.

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